Repeating yourself with repeating decimals

      When a decimal repeats itself, you can always find the fraction that corresponds to the decimal. In this chapter, I only cover the decimals that show every digit repeating.

      To change a repeating decimal (in which every digit is part of the repeated pattern) into its corresponding fraction, write the repeating digits in the numerator of a fraction and, in the denominator, as many 9’s as there are repeating digits. Reduce the fraction if necessary.

      A. . The three repeating digits are 126. Placing the 126 over a number with three 9’s, you reduce by dividing the numerator and denominator by 9.

      Q. Write the decimal as a fraction in lowest terms.

      A. The six repeating digits are put over six 9’s. Reducing the fraction takes a few divisions. The common factors of the numerator and denominator are 11, 13, 27, and 37. When completely reduced, you have .

      44 Change to a decimal.

45 Change to a decimal.

      46 Change to a decimal.

      47 Change 0.45 to a fraction.

      48 Change to a fraction.

      49 Change to a fraction.

50 Round the decimal 4.172797 to the nearest thousandth.

Practice Question Answers and Explanations

      1 . First, divide the 29 by 8. The number 8 divides 29 three times with a remainder of 5.

      2 . Multiply the 4 and 9 and then add the 5, which equals 41. Then write the fraction with this result in the numerator and the 9 in the denominator.

      3 . Multiply the 4 times 100 and add the 7. Put the sum over 100.

      4 . Ignore the negative sign at first; you don’t want it involved in the computation. First multiply the 2 times 13 to get 26. Add the 1 to get 27. You have 27 in the numerator, 13 in the denominator, and now you put the negative sign in front.

      5 . The number 11 divides 402 a total of 36 times with 6 left over. The 36 goes in front, with the 6 in the numerator and 11 in the denominator. This example makes it especially apparent that the mixed number is more understandable.

      6 . Divide 7 into 19, for a quotient of 2. The remainder 5 goes in the numerator. Put the negative sign in front of the 2.

      7 . To get 28 in the denominator, multiply 7 by 4:

      8 . Multiply both the numerator and denominator by 5:

      9 . The number 4 is the greatest common divisor of 16 and 60 because and . So multiply by to get

      Or, if you prefer, divide both the numerator and denominator by 4:

      10 . The largest common divisor of 63 and 84 is 21, because and . So

But let’s say you do this in two steps — both dividing by a common factor.

      You see that both 63 and 84 are divisible by 7. So divide both the numerator

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